rt.representation theory - Functions on hyperbolic space and modular curves

The decomposition of $L^{2}left(S^{2}right)$ under $SOleft(3,mathbb{R}right)$ is well-known.

Focus now on the hyperbolic plane $H$ presented as the quotient $SLleft(2,mathbb{R}right)/SOleft(2,mathbb{R}right)$. It is non-compact, therefore my understanding is that infinite-dimensional
representations of $SLleft(2,mathbb{R}right)$ will appear in the decomposition of $L^{2}left(Hright)$.

(a) Is there an algebraic part of the spectrum and does it have a description
similar to the one in $L^{2}left(S^{2}right)$?

(b) How to classify the $SLleft(2,mathbb{R}right)$ representations and what is the whole spectrum?

(c) Consider $X_{0}left(1right):=SLleft(2,mathbb{Z}right)setminus H$. How does $L^{2}left(X_{0}left(1right)right)$ decompose?

(d) The same for $X_{0}left(Nright):=Gamma_{0}left(Nright)/H$. How does $L^{2}left(X_{0}left(Nright)right)$ decompose?